how to extend an object just on one side along his own edges?
You can still use edge slide to do this.

Select the end face (or its edges or vertices)
GG.. edge slide
Draw the face inwards a little way, making the piece shorter as you described in your post (this establishes the slide directions)...
Now hold down Alt, or tap C releasing the clamp. You should see rays extending the longitudinal edges.
As long as you are holding Alt, you can slide the end face's vertices anywhere you like along those rays.


Unfortunately, snap doesn't work with GG. If you wanted that, too, I think you would have to break the operation up, separately moving components along Custom Orientations derived from the longitudinal edges.